Topology Optimization Procedure Research Articles

Practical Design Guidelines for Topology Optimization of Flexible Mechanisms: A Comparison between Weakly Coupled Methods

Industrial robots are complex systems, as they require the integration of several sub-assemblies to perform accurate operations. Moreover, they may experience remarkable dynamic actions due to high kinematic requirements, which are necessary to obtain reduced cycle times. The dynamic design of industrial robots can therefore be demanding, since the single structural component can induce an impact both in the design phase (development strategy and computational time) and at the machine level (global stiffness and natural frequencies). To this end, the present paper proposes first a topology optimization procedure based on the Equivalent Static Loads (ESL) method that integrates flexible multibody simulation outputs. The same procedure also foresees an intermediate static reduction to reduce and to precisely define the application points of the ESL. Secondly, an optimization procedure based on the Quasi-Static Loads (QSL) method integrating flexible multibody simulation outputs is proposed as well. The objective is to carry out a comparison between the two methods and consequently evaluate the benefits and drawbacks of the two. In the end, practical guidelines regarding the selection and application of the two methods are also provided to the reader.

An anisotropic topology optimization procedure for continuous fiber reinforced polymer structures with biaxial fiber layout for improved intersection point design

Fiber placement technologies allow for manufacturing of complex-shaped composite parts with load-adapted fiber orientation. For design of corresponding structures, topology optimization is well-established. However, anisotropic topology optimization approaches are often limited to unidirectional fiber orientations. In truss-like topology optimized structures, the intersections between the individual trusses are subject to multiaxial stress states. For those, a unidirectional fiber orientation is not appropriate and may not be suitable for manufacturing.In this work we propose a topology optimization procedure that allows for unidirectional but also biaxial fiber orientations. The choice between unidirectional and biaxial fiber layout is based on the stress state within the finite elements. For the biaxial fiber layout, the first fiber angle is aligned with the maximum absolute principal stress direction, while the second fiber orientation is calculated using composite net theory.Results show that considering biaxial fiber orientations leads to fiber orientations that can be better manufactured by fiber placement technologies and are considered suitable for automatic derivation of manufacturing-tolerant placement paths. Furthermore, the stress state at intersections is improved reducing the probability of inter fiber failure. In addition, consideration of biaxial fiber orientations could increase the stiffness at constant weight compared to purely unidirectional topology optimization.

Efficient inverse-designed structural infill for complex engineering structures

Inverse design of high-resolution and fine-detailed 3D lightweight mechanical structures is notoriously expensive due to the need for vast computational resources and the use of very fine-scaled complex meshes. Furthermore, in designing for additive manufacturing, infill is often neglected as a component of the optimized structure. In this paper, both concerns are addressed using a so-called de-homogenization topology optimization procedure on complex engineering structures discretized by 3D unstructured hexahedrals.Using a rectangular-hole microstructure (reminiscent to the stiffness optimal orthogonal rank-3 multi-scale) as a base material for the multi-scale optimization, a coarse-scale optimized geometry can be obtained using homogenization-based topology optimization. Due to the microstructure periodicity, this coarse-scale geometry can be up-sampled to a fine single-scale physical geometry with optimized infill, with only a minor loss in structural performance and at a fraction of the cost of a fine-scale solution. The upsampling on 3D unstructured grids is achieved through stream surface tracing, aligning with the optimized local orientations. The periodicity of the physical geometry can be tuned, such that the material serves as a structural component and also as an efficient infill for additive manufacturing designs.The method is demonstrated through three examples of varying geometrical complexity. It achieves comparable structural performance to “brute force” state-of-the-art methods but stands out for its significant computational time reduction. By allowing multiple active layers, the mapped solution becomes more mechanically stable, leading to an increased critical buckling load factor without additional computational expense. The control of active layers also provides direct control over the internal structure, i.e., infill, ensuring that the infill is incorporated as a structural component and enhancing the manufacturability of the de-homogenization procedure. Furthermore, the proposed approach exhibits promising results, achieving volume fractions and weighted compliance values within 5% of the large-scale SIMP model, while demonstrating a computational efficiency improvement ranging from 10 times to over 250 times.

Multiscale topology optimization framework for natural frequency maximization of multi-morphology lattice structures

This paper proposes a multiscale topology optimization framework for maximizing the natural frequency of multi-morphology lattice structures (MMLSs). The proposed framework addresses the challenges of computational efficiency, design space, numerical convergence, and compatibility between adjacent microstructures in multiscale topology optimization for natural frequency problems. The macrostructural topology, the morphologies categories, distribution regions, volume fractions of different lattice materials (LMs), and the relative densities of lattice unit cells (LUCs) are simultaneously optimized to enhance the structural natural frequency. Specifically, level set functions are utilized to generate prototype LUCs, enabling obtaining graded LUCs by configuration interpolation. Multi-morphology LUCs with smooth characteristics are achieved using the Kriging-assisted morphological post-process and sigmoid function (SF) based hybrid transition strategy. Kriging metamodel-assisted Uniform Multiphase Materials Interpolation (KUMMI) schemes are constructed for the mechanical properties estimation of macro elements to promote the multiscale topology optimization procedure. Distinct processing methods for LUCs' elasticity tensor and density are incorporated within the KUMMI schemes, by which the local mode problem is avoided. The objective order natural frequency is accurately optimized with the modal assurance criterion (MAC) based mode-tracking technique. Numerical examples demonstrate that the developed design framework can efficiently maximize the natural frequency of MMLSs, while also ensuring microstructural connectivity.

Full-waveform inversion using level set and cut elements for sharp-interface problems

This paper proposes a topology optimization procedure to reconstruct sharp interfaces of salt bodies in a background medium with known properties. The reconstruction technique uses the time-domain acoustic wave equation that provides the pressure field to minimize a data misfit objective function. The salt-background distinction is described by the level set-cut element approach proposed by Andreasen et al. (2020) [1]. This approach preserves the abrupt transition of properties across interfaces and provides accurate solutions to the problem. The nodal values of the level set function are parameterized as an explicit function of the design variables, and a “discretize (space)-then-derive”approach is implemented consistently with the discretized optimization problem. This setting allows for solving the minimization problem with mathematical programming methods; we use the conjugate gradient to update the level set shape-based reconstruction. A set of numerical examples demonstrate that the optimization algorithm is quite robust and is able to identify complex geometries of salt bodies.

Modeling, testing, and optimization of novel lattice structures for enhanced mechanical performance

Cellular materials have drawn increasing interest in numerous applications due to their promising specific stiffness, strength and energy absorption capacity. In this work, a variety of rather novel lattice topologies pertinent to additive manufacturing are derived and examined. A number of these are derived by free-domain and constrained domain topology optimization procedures, while others are inspired by the triply periodic minimum surface (TPMS) sheet-based topologies. The topology optimization module utilized a single objective function of minimizing strain energy under linear elastic conditions. A total of fifteen different lattice topologies are investigated numerically, including both novel and conventional topologies (e.g. strut-based lattices) and their effective elastic properties are determined with respect to relative density through finite element analysis (FEA). Based on the preliminary FEA results, a number of these topologies are selected of which tessellated lattice structures are fabricated through laser powder bed fusion (LPBF) additive manufacturing technique out of Nylon thermoplastic material. The tessellated lattice structures are experimentally tested in compression and their mechanical performance, including uniaxial modulus, yield strength, and energy absorption capacity (EAC), is assessed. FEA simulations have been conducted using an elastic-plastic constitutive model for the Nylon base material. Both the experimental and numerical results reveal that the mechanical performance of the novel tube-based TPMS lattice P-100 and the combined loading (CL) topology derived through free-domain topology optimization surpasses all other topologies. P-100 uses a primitive TPMS with equal-length tubular connections in each direction, where the tubular length percentage compared to the primitive lattice size is 100%, while CL lattice topology is a free domain topology optimized under compressive loads on the centers of faces, edges, and vertices toward the center. The innovative lattice topologies proposed in the current study, particularly the P-100 and CL topologies, can become crucial in applications where it is necessary to improve the energy absorption capacity, such as sandwich panel cores, supports, and infills for 3D printed components.

Topology optimization approach for functionally graded metamaterial components based on homogenization of mechanical variables

The micron resolutions that 3D printers are reaching allow for a change of paradigm in the design of components, optimizing not only the topology of the component, but also the microstructure of the material, which can adopt different objectives at different locations in the component, reaching different mechanical properties and different optimal microstructures at those locations. This constitutes a flip in the design objectives to focus on tailored material work, aimed either at local objectives or at component-wide ones, instead of pursuing homogeneous material properties for global objectives. For this new concurrent structure-material design leading to optimized components made of functionally graded (FG) metamaterials, new topology optimization (TO) procedures are needed in which the objective is to obtain component-wise mechanical variables or distributions of these variables, rather than global component objectives as compliance, which is a typical objective of classical TO approaches.In this work we present a new TO approach aimed at FG metamaterial design pursuing homogeneous mechanical variables. We focus on homogenization of deformation level along the component by minimizing its standard deviation across the domain. The proposed formulation can be naturally generalized to anisotropy, nonlinear behavior, and other optimization objectives. Convergence to a suitable optimized solution, with a distribution of material properties resulting in a homogeneously deformed component, is obtained. These FG material properties may be obtained from local metamaterial design. Classical material/void configurations may also be obtained following our procedure, and results for such cases can be compared qualitatively to the Solid Isotropic Material with Penalization (SIMP) approach.

Cross-scale optimization of advanced materials for micro and nano structures based on strain gradient theory

Recently developed cross-scale optimization methods are mainly the macro equivalent calculation of performances based on the traditional homogenization method. However, the traditional homogenization is limited to the classic continuum of Cauchy–Boltzmann. Therefore, it is inadequate to interpret the size dependence of the optimal result. Hence, a new cross-scale optimization is proposed based on Wei–Hutchinson strain gradient theory by employing the non-local homogenization model, which could describe and explain the size dependence during optimization process when considering micro structures. The topological optimization procedure simultaneously has the ability of coupled computing by using subdomain parameterized coarse meshes. The numerical computations involved in the entire model can be solved in one iteration, which helps to eliminate mesh dependencies and greatly reduce the computation time. It is shown that the final stiffness of the optimized periodic structure can be significantly increased by considering the strain gradient theory compared with the classic homogenization scheme in the process of cross-scale optimization.

Multiscale topology optimization of pelvic bone for combined walking and running gait cycles

We propose a multiscale topology optimization procedure of pelvic bone using weighted compliance minimization. In macroscale optimization, a level set-based method is used, which gives a binary structure. In microscale optimization, cubic lattice-based homogenization is done while keeping the global geometry fixed. For the macroscale, a volume constraint equal to the volume of the pelvic bone is imposed, whereas, for the microscale, a mass constraint equal to the mass of the pelvic bone is imposed. The optimal geometries are compared with pelvic bone using different metrics and show good similarity with the same. Designed geometries are additively manufactured and experimentally tested for stiffness.

Multi-objective topology optimization of periodic pile barriers with ambient body waves isolation performance

The attenuation zones (AZs) of periodic pile barriers can be used to mitigate ambient vibration. To identify pile structures with optimal vibration isolation performance and minimal concrete consumption, this study adopts a multi-objective topology optimization (MOTO) procedure to optimize the periodic pile units for the isolation of plane waves. The optimized pile structures are presented, and the most cost-effective one is suggested. The effects of the AZ orders, the elastic modulus and the density of soil on the optimization results are investigated. By establishing numerical models, the vibration reduction performance of the optimized pile barriers in the frequency domain and time domain is verified.

A level set-based interface-enriched topology optimization for the design of phononic crystals with smooth boundaries

Phononic crystals can be designed to show band gaps—ranges of frequencies whose propagation is strongly attenuated in the material. In essence, their working principle is based on destructive interference of waves reflecting from the periodic arrangement of material interfaces (i.e., Bragg scattering). Consequently, capturing accurately the behavior at material interfaces requires appropriate numerical modeling and computational design techniques. However, the commonly used density-based representation in popular topology optimization methods results in a diffuse staircased boundary. The heavily refined finite element meshes required to compensate for this boundary description results in exceedingly large and expensive optimization problems. In this paper, we demonstrate the adverse effect of the density-based boundary description. Furthermore, we propose a level set-based topology optimization procedure with an enriched finite element method that shows improved performance when compared to the density-based approach.

Extending the usability of the force-flow based topology optimization to the process of generative design

Topology optimization and Generative Design are two methods to create volume or stiffness optimized parts and they are used more frequently nowadays. However, the specific methods are often not well described and the connection between these two is not clearly explained. In this article, a force-flow based topology optimization process has been explained in detail and extended with a function to be able to use as a Generative Design tool. The proposed algorithm has been tested on three 2D shapes and the effectiveness was evaluated. This work clarifies the vague description of theoretical solutions by presenting in detail the operation of the algorithm and bridging the lack of information that exists between the shape or topology optimization procedure and the generative design solution.

Addressing topology optimization with overhang constraints for structures subjected to self-weight loads

This paper investigates the topology optimization of structures subjected to self-weight loads with self-supporting constraints for additive manufacturing. The integration of topology optimization procedures and additive manufacturing techniques can make the most of their advantages, and there is significant interest today in integrating both approaches. Imposing overhang constraints in topology optimization has been addressed, but primarily for classical topology optimization problems with fixed external loads, not design-dependent loads. This work combines an effective numerical procedure for contour evaluation with a modified version of the power-law model for low densities to eliminate the problems that arise when self-weight loads are considered. The overhang edge detection is based on the Smallest Univalue Segment Assimilating Nucleus (SUSAN) method, and a variable mask size technique is used to avoid eventual dripping problems. The proposed constraint function evaluates the overhang globally and allows control of the formation of unsupported contours for maximum stiffness design problems when self-weight loads are present. Several numerical experiments demonstrate the proposed method's effectiveness and robustness.

SVR-TO-APMA: Hybrid efficient modelling and topology framework for stable topology optimization with accelerated performance measure approach

The topology design under uncertainties is an acceptable framework for providing a safe design and optimum configuration of structures. In this regard, the accuracy, efficiency, and robustness of the analytical topology optimization coupled with reliability analysis are the main efforts for reliability-based topology optimization (RBTO) of continuous structures. In the present investigation, a hybrid RBTO approach is developed for robust iterative formulation of reliability loop and an effective and simple optimization approach for topology loop. A multi-level set analytical framework is proposed based on the performance measure approach to seek the optimum results under multi uncertainties in size, mechanical properties, and loads of structures. The considered topology optimization procedure to derive the optimal layout of the structure which is called moving iso-surface threshold (MIST) is a powerful and enhanced topology optimization method in which, unlike other optimization methods, it has the capability of having no direct sensitivity analysis making it easier and simple. The accelerated first-order reliability method is introduced for searching the most probability points in the reliability loop of the RBTO model. The proposed reliability method is developed based on a stable iterative formulation. For reducing the computational burden of the topology optimization loop, the machine learning approach given from the support vector regression (SVR) is applied for estimating the probabilistic constraints in the RBTO model. Four examples of 2D, and 3D compliance, 2D complaint mechanism problem, and frequency maximization are presented for validation and highlighting the abilities in having the most accuracy and safety levels of the proposed hybrid framework named SVR-TO-APMA. The results indicate that the hyperparameters of SVR are directly affected by the accuracy of RBTO results. The stability of the RBTO model is shown by the accelerated PMA and the optimum layout is different from topology optimization in comparison with the RBTO results for 3D compliance and frequency maximization problems. The optimum design with uncertainties is increased by about 40%–90% with a reliability index of 3 compared to TO results for the studied example.

Topology optimization of nonlinear flexoelectric structures

Recent advances in nanotechnology have allowed for the manufacturing of nanostructures and nanodevices with optimized topologies that outperforms their traditional counterparts based on simple geometry in terms of efficiency and function. In this work, a novel nonlinear topology optimization procedure is developed to design optimal layouts of flexoelectric structures undergoing large displacement. The optimal material distribution is determined by optimizing the energy conversion efficiency. Two material properties such as the elasticity coefficient and the permittivity coefficient are interpolated through the solid isotropic material with a penalization approach via an energy interpolation scheme to overcome the numerical instability. Obtained results have revealed that accounting for the geometric nonlinearity leads to a different final optimal topology with a better energy converting efficiency as compared to the linear model. We also found the significant influences of size effects on the optimized structure emphasizing the importance of nonlocal elastic behavior in characterizing and designing micro-structures and flexoelectricity.

Integration of hydrogen storage and heat storage in thermochemical reactors enhanced with optimized topological structures: Charging process

The low density for hydrogen storage can be solved by metal hydrides, and the energy loss for hydrogen storage with metal hydrides can be recovered by the combination of metal hydrides (Mg/MgH2) with thermochemical heat storage materials (MgO/Mg(OH)2) under the different reaction temperatures. However, the poor heat conduction of thermochemical materials limits the heat transfer rate and the implementation of this technology. In this study, an accurate theoretical model of hydrogen charging process in the hydrogen storage reactor assisted with heat storage was developed to accommodate the operating pressure. A novel method to overcome the poor heat transfer problem was proposed by topology optimization of high thermal conductivity fins for matching the heat transfer and reaction rate in the hydrogen storage side with those in the heat storage side. Increasing the average temperature of MgO/Mg(OH)2 was considered for the formulation of the topology optimization problem, and a density-based approach was used. The influence of the conductivity of materials on the topology optimization was discussed. The outcomes displayed that higher thermal conductivity accommodated more heat transfer directions, resulting in a distinct branch structure. Furthermore, the obtained topologies were geometrically reconstructed and evaluated during the hydrogen charging process. Results revealed that the topology optimized geometry had outstanding heat transfer performance, with an obvious improvement in terms of hydrogen charging time over the reactor without fins. This work can provide the theoretical basis for thermochemical conversion processes of hydrogen storage and heat storage, the implementation procedure for topology optimization, and the design guidance of thermochemical reactors for hydrogen storage assisted with heat storage.

Topology Optimization of Perforated Steel Strip in Composite Structures

A shear connection used in composite structures consisting of perforated steel strips has been under investigation for almost forty years. During this time, several suitable shapes of perforated strips were investigated. Many of them provide sufficient behaviour in the transmission of longitudinal shear between a steel beam and a concrete slab. Instead of developing a completely new shape of the recesses in the perforated strip with unknown behaviour under static and dynamic loading, a steel strip already examined was chosen as the reference type for this study. Subsequently, a topology optimization procedure was performed to analyse the possibilities of shape customization to improve behaviour by increasing not only the resistance of the entire shear connection but also the ductility of the steel strips. Based on the results of the verifications, this procedure showed possibilities for improvement but also deterioration of the properties of the examined perforated steel strips, as well as other types of strips. Furthermore, by appropriate rearrangement of the reinforcement bars, the increase in resistance value can be significantly higher.

Band-gap design of reconfigurable phononic crystals with joint optimization

In this article, a joint topology optimization procedure is executed to design phononic crystals (PnCs) with a maximized band gap. First, the band gap of two-dimensional PnCs is calculated using a finite element model under the plane strain assumption for easy and efficient optimization. In the numerical computation of the band gaps, the band structures of in-plane and out-of-plane modes are analyzed and verified using COMSOL. Then, the density-based and feature-driven methods are sequentially used to design the unit cell of the PnCs with single-phase material. A stiffness constraint is added to ensure the continuity of the material inside the unit cell, and a boundary constraint guarantees continuity amongst the unit cells. The optimization problem is solved through the Method of Moving Asymptotes (MMA), and reconfigurable manufacturing PnC is obtained via joint topology optimization. Lastly, the band structures and vibration modes of the reconstructed model are analyzed. Research results show that the optimal PnCs generate broadband vibrations by localized resonance and have negative refraction characteristics. This research also confirms that the joint topology optimization is an efficient approach for designing novel elastic metamaterials.

Topology optimization of structures with infill-supported enclosed voids for additive manufacturing

This paper proposes a novel topology optimization procedure for designing structures with infill-supported enclosed voids for additive manufacturing (AM). In such structures, the open and enclosed regions are separately treated, where the open regions are the standard voids but the enclosed ones are filled by porous materials. The applied porous infill can work as supports in the AM process, but at the same time be part of the final structures and will never be removed. In this way, the difficulty of removing the supports in enclosed voids can be naturally avoided. However, whether the enclosed voids should be removed or kept if the porous material must be filled inside becomes a challenging problem, and thus this work develops a rational procedure to automatically determine it, and optimize the layout of these kept enclosed voids with infill. In the proposed procedure, a nonlinear virtual temperature (N-VTM) filter is utilized to separate the enclosed voids and open regions. Then, by combining the N-VTM filter, PDE filter and projection, a novel interpolation for structures with infill-supported enclosed voids is proposed, which at the same time can be used to provide structures without enclosed voids. The robust topology optimization formulation is applied to enhance the performance of the proposed procedure, and the sensitivity equations are deduced based on the adjoint method. Several numerical and experimental examples in two and three dimensions illustrate the effectiveness of the procedure. The results show that the procedure is capable of automatically determining whether to directly eliminate an enclosed void or to use infill to support it.

Two-level method part-scale thermal analysis of laser powder bed fusion additive manufacturing

Numerical simulations of a complete laser powder bed fusion (LPBF) additive manufacturing (AM) process are extremely challenging, or even impossible, to achieve without a radical model reduction of the complex physical phenomena occurring during the process. However, even when we adopt a reduced model with simplified physics, the complex geometries of parts usually produced by the LPBF AM processes make this kind of analysis computationally expensive. In fact, small geometrical features—which might be generated when the part is designed following the principle of the so-called design for AM, for instance, by means of topology optimization procedures—often require complex conformal meshes. Immersed boundary methods offer an alternative to deal with this kind of complexity, without requiring complicated meshing strategies. The two-level method lies within this family of numerical methods and presents a flexible tool to deal with multi-scale problems. In this contribution, we apply a modified version of the recently introduced two-level method to part-scale thermal analysis of LPBF manufactured components. We first validate the proposed part-scale model with respect to experimental measurements from the literature. Then, we apply the presented numerical framework to simulate a complete LPBF process of a topologically optimized structure, showing the capability of the method to easily deal with complex geometrical features.